Question

find all the zeros of the polynomial 2x⁴ + 7x - 19x² - 14x + 30, if two of its zeroes are √2 and -√2
can anyone plss explain in notebook??​

Answer 1

$\huge\purple{\boxed{\underline{ANSWER}}}</p><p>$

GIVEN THAT:-

$&#10154$ polynomial = 2x⁴ + 7x3 - 19x² - 14x + 30 =

$&#10154$ two of its zeroes are √2 and -√2

EXPLANATION:-

Given that √2 and -√2 are two zeros of this polynomial so,

$&#10154$. (x - √2) and ( x + √2) will be tow factor of this polynomial

as well as

$&#10154$ (x-√2)( x+√2) = ( x² - 2 ) will be also the factor of Given polynomial .

Now dividing the polynomial with ( x² - 2 )

$&#10230 \: \: \frac{2 {x}^{4} + 7 {x}^{3} - 19 {x}^{2} - 14x + 30 }{ {x}^{2} - 2 } \\ \\ &#10230 \: \: 2 {x}^{2} + 7x - 15 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Now we get a quadratic equation

so

$&#10230 \: \: 2 {x}^{2} + 7x - 15 \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ &#10230 \: \: 2 {x}^{2} + 10x - 3x - 15 \\ \\ &#10230 \: \: 2x(x + 5) - 3(x + 5) \\ \\ &#10230 \: \: (x + 5)(2x - 3) \: \: \: \: \: \: \: \: \: \: \\ \\ &#10230 \: \: x = - 5 \: and \: \frac{3}{2 } \: \: \: \: \: \: \: \: \: \: \: \: \:$

Now the all zeros of Given polynomial

√2 , - √2 , -5 and 3/2 .

Thanks